Munich Personal RePEc Archive

The Performance of UK Securitized

Subprime Mortgage Debt: ‘Idiosyncratic’

Behaviour or Mortgage Design?

Lanot, Gauthier and Leece, David

Keele Management School (Economics), Keele University,Manchester Metropolitan University Business School

15 November 2010

Online at https://mpra.ub.uni-muenchen.de/27137/

MPRA Paper No. 27137, posted 01 Dec 2010 14:26 UTC

1 The Performance of UK Securitized Subprime Mortgage Debt: ‘Idiosyncratic’

Behaviour or Mortgage Design? Professor David Leece (contact author), Professor of Financial Studies, Manchester

Metropolitan University Business School, Aytoun Building, Aytoun Street, Manchester,

M13GH, United Kingdom. Contact: d.leece@mmu.ac.uk or tel. +44(0)161 247 3742

Professor Gauthier Lanot, Professor of Empirical Economics, Keele Management School,

Keele University, Keele, Staffordshire, ST5 5BG, United Kingdom.

Contact: g.lanot@keele.econ.co.uk or tel. +44 (0)1782 733102

2

The Performance of UK Securitized Subprime Mortgage Debt: ‘Idiosyncratic’ Behaviour or

Mortgage Design?

Professor David Leece, Manchester Metropolitan University Business School, UK

Professor Gauthier Lanot, Keele University Management School , Keele, UK

Abstract

The research estimates a competing risk model of mortgage terminations on a sample of UK

securitized subprime mortgages. We consider whether the variety of mortgage contracts

that were securitized explains the performance of subprime securities and their supposed

‘idiosyncratic’ behaviour. The methodological advance is the use of a general, flexible

modelling of unobserved heterogeneity over several dimensions, controlling for both

selection issues involving mortgage choice and dynamic selection over time. We conclude

that securities consisting of subprime loans can be given meaningful valuations on bank

balance sheets if the performance of the different types of loans can be better understood.

JEL Codes: G21 C13 C25 C51 D10 D14 E44

Key Words: Subprime mortgages, unobserved heterogeneity, loan performance,

securitisation

3 The Performance of UK Securitized Subprime Mortgage Debt: ‘Idiosyncratic’

Behaviour or Mortgage Design?

1. INTRODUCTION

The paper reports the results of estimating a model of default and prepayment behaviour

with competing risk and unobserved heterogeneity, using a sample of securitized United

Kingdom (UK) subprime mortgages originated before the financial difficulties of 2007 and

subsequently held by a global investment bank. We consider whether the variety of types of

mortgage that were securitized including discounted debt, fixed mortgage interest rates

(typically fixed for two or three years), and self certified (that is low or zero documentation)

mortgages significantly influenced the default and prepayment behaviour of securitized

subprime loans. Given the conventional wisdom that these securities are idiosyncratic and

difficult to value this is an important question. Is the variation in loan performance between

pools of subprime mortgage debt not only attributable to the contracts contained in the

securities, but also the selectivity surrounding such choices? The broader puzzle is why such

a variety of contracts were securitized and whether such contract variety is conducive to

securitisation?

The motivation for the research is the recent credit crisis and its legacy. The ‘aggressive’

extension of mortgage lending to subprime borrowers leading to losses on pools of

subprime debt are seen as the proximate causes of the global credit crunch that began in

2007. The analysis of the loan performance of quits from mortgage pools of United States

(US) prime borrowers, and to some extent subprime mortgage holders, are reported

elsewhere (Alexander et al, 2002; Chomsisengphet and Pennington Cross, 2004; Courchane

et al, 2004; Cowan and Cowan, 2004; Danis and Pennington Cross, 2008; Stephens and

4 Quigars, 2008) and report a variety of results concerning the effects of different contract

design on loan performance, and the influence of variables representing the value of

embedded options in a mortgage contract. Our work adds to this literature, covering

mortgages originated and held in a more recent time period (2001-2008). This sample

period is of interest in its own right, since a conventional explanation for the recent financial

crisis orginates with the performance of subprime loans. An analysis of subprime loan

performance is important for better understanding the nature and causes of the financial

crisis.

The methodology of the research allows an extension of the modelling of behaviour beyond

the data typically available to the securitisers. Unobserved individual influences, that is

unobserved heterogeneity, would appear to be an obvious source of variation in

loan/security performance and the basis of idiosyncratic behaviour. This paper adds to

current research by introducing a general and flexible modelling of this heterogeneity over

the relevant two dimensions of default and prepayment behaviour. This approach provides

a framework capturing both the selection issues arising from initial mortgage choices, and

the dynamic selection effects arising from changes in the population of individuals who

remain in the pool of mortgages at each discrete point in time.

To our knowledge, there is no econometric work on subprime loan performance for the

United Kingdom which is comparable with that undertaken for the United States mortgage

market. This is a major omission given the extent to which UK mortgages were securitized,

together with the variety of and distinctive contractual features of UK housing debt (see

Leece, 2004; Miles, 2005). The Bank of England estimated that the stock of outstanding non

conforming securitized debt in the UK to be £39billion (Bank of England, 2008). An

5 understanding of the factors underpinning UK subprime loan performance, and the

problematic nature of pricing risk on these securities, contributes to the debates on

valuation; on future requirements of any reinvigorated securitized mortgage market; on

mortgage contract design; on risk management; and on financial regulation.

The paper begins with a review of the academic literature on the default and prepayment

behaviour of mortgage loans and subprime loans where applicable. This is followed by an

outline of the econometric methodology and the modelling approach to unobserved

heterogeneity. The sample and the empirical specification of the model are discussed in the

section which follows. Parameter estimates are then reported and analysed.

2. DEFAULT AND PREPAYMENT

This section of the paper presents the key theoretical and empirical approaches adopted in

the mortgage loan performance literature and positions the research in relation to that

work. The discussion facilitates the identification of the key influences upon default and

prepayment behaviour which informs the empirical model that follows.

The research into subprime loan performance emphasizes: the influence of contract

features such as prepayment penalties and reduced documentation upon the probability of

foreclosure (Quercia, Stegman and Davis, 2005; Rose, 2008); the effect on default rates of

originating loans from third parties (Alexander et al, 2002; Pennington Cross, 2003); and

how default rates vary by loan classification (Cowan and Cowan, 2004). Econometric

specifications and results in the research of the subprime market tend to reflect the

research into the behaviour of prime mortgage loans, with the co-variates for subprime

debt having larger marginal effects on the default and/or repayment probabilities

6 (Pennington Cross and Chomsisengphet, 2007). Thus the empirical modelling also draws

upon the wider mortgage loan performance literature.

Option Theoretic and Empirical Prepayment Models

The literature on the options embedded in mortgage contracts and their impact upon loan

performance is well established, both theoretically and empirically (Kau et al, 1993; Deng et

al, 2000; Ambrose and LaCour-Little, 2001; Ambrose and Sanders, 2003). The possibility of

defaulting on a loan is treated as a put option (selling the house back) while prepayment is

considered as a call option (buying back the mortgage). The analysis is to some extent

United States specific, for example in the UK the borrower retains liability for the

outstanding mortgage debt on default (Leece, 2004), i.e. the put option may not be so

valuable. The prepayment option also applies more readily to long term fixed interest rate

mortgages more typical of the US (Leece, 2004). However, short term fixed rates and

periods were the interest rate is discounted, but eventually reverts to an higher rate,

provide boundary conditions for valuing the call option (Kau, 1993).

Previous work suggests that borrowers do not always take systematic advantage of the

embedded options they hold, such as not prepaying when favourable alternative contracts

are available (the call option is in the money) or not defaulting when the put option is well

into the money. This has led to several developments. One is to estimate empirical

prepayment models that recognise the importance of exogenous effects (surprises) on

default and prepayment behaviour, for example the effect of payment shocks (Quigley and

Van Order, 1990, 1995; Archer and Ling, 1993). In particular some work has focused on

studying loan level data where borrower characteristics can be analysed. Furthermore, the

7 literature has further emphasized the role of unobserved heterogeneity among borrowers

(Deng and Quigley, 2002; Alexander et al, 2002).

The majority of recent empirical studies of mortgage loan performance now incorporate

modelling of both the embedded options in mortgage contracts and variables typical of

empirical prepayment models. There is also a recognition that the embedded options

represent a competing risk in that the exercise of one option precludes the exercise of the

other (Deng et al, 2000; Lambrecht et al, 2006). The research reported in this paper uses

loan level data and estimates an empirical model of mortgage default and prepayment

which incorporates both an option theoretic specification and includes variables that impact

upon affordability, or reflect exogeneous shocks. The study is unusual in incorporating a

wide variety of mortgage contracts in the sample which makes explicit issues regarding

selection bias which have not been treated in previous work looking at single types of

contract.

Mortgage Design

The mortgage contracts studied in the US are typically fixed rate mortgages with the interest

rate fixed for 15 or 30 years, and adjustable rate mortgages (ARM) where the rate of

interest changes annually. This compares to UK mortgage contracts where the majority have

interest rates fixed for one to three years or the interest rate changes at irregular periods (a

variable rate mortgage). The UK one year fixed rate contracts are similar to the US

adjustable rate mortgage and the two to three year interest rate fixes are equivalent to the

so called ‘hybrid mortgage’ in the US (see Ambrose et al, 2005). The details of these

contracts can have an impact upon prepayment and/or default behaviour, for example

prepayment penalties (Pereira et al, 2002).

8 There is evidence for the United States that adjustable rate mortgages prepay at a faster

rate than fixed rate mortgages (Ambrose and LaCourLittle, 2001). Households holding

adjustable rate mortgages may be more mobile than those with fixed rate contracts and will

start refinancing after a short period of time (Brueckner, 1995). Borrowers choosing a

discounted mortgage may chase new (better) deals. Thirdly ARM borrowers may refinance

into fixed rate mortgages on the reset date, depending upon interest rate expectations.

Hence discounted mortgage holders will tend to have a higher likelihood of prepayment

than fixed rate mortgage holders. The payment shock which arises around the date of

adjustment of ARM interest rates might induce a higher level of mortgage defaults

(Ambrose et al, 2005). A similar phenomenon takes place with UK short term fixed rate

debt.

The mortgage contracts featuring in the current research involve self-certification,

discounting, fixed rate contracts, and dates at which the interest rate on the contract

reverts from a favourable rate of interest back to an higher index rate (typically London

Interbank Offered Rate, LIBOR) plus margin (reset dates). The real estate economics and

finance literature has examined the effects of these different aspects of contract design on

the performance of mortgage loans (Phillips et al, 1996; Vanderhoff, 1996; Green and

Shilling, 1997; Ambrose and LaCour-Little, 2001). The majority of papers have considered

the effect of discounting the initial interest rate (teaser rates). The empirical results have

been mixed and contradictory. Later work should be credited with the use of a competing

risk framework and controlling for unobserved heterogeneity. Ambrose and LaCour-Little

(2001) apply this methodology and find a significant increase in prepayments at adjustment

dates.

9 A further key feature of recent mortgage contracts has been low or zero documentation in

the US and its equivalent the self certified mortgage in the UK1. This relaxed approach to

mortgage underwriting attenuates or overrides prudential lending criteria and introduces

information asymmetry, with the lender knowing less about the borrower’s ability to pay

and likelihood of default. There may be substantial adverse selection and borrowers may

exhibit opportunistic behaviour (Brueckner, 2000; Leece, 2004). As a consequence self-

certification will have a positive effect on the likelihood of defaulting. There is very little

research in this area, an exception is Rose (2008) who estimates a multinomial logit model

with unobserved heterogeneity using securitized subprime loans for the Chicago

Metropolitan area from January 1999 and up to mid 20032. Rose finds that the effects of

variables on foreclosure depend upon loan features such as the level of documentation.

A key motivation for researching the impact of contract designs on loan performance is the

question of which loans should be securitized and how such securities would be valued.

Until recently, adjustable rate and other mortgage designs were less likely to have been

securitized than long term fixed rate contracts (Ambrose and La-Cour Little, 2001).

Furthermore, loan contracts which exhibit complex and less predictable default and

repayment rates may be less suitable for securitisation. Research to date has typically

controlled for different contract designs by analysing loans of a given type (typically long

term fixed or adjustable rate). Even analysing one type of loan raises issues regarding 1 Self-certified mortgages are designed for self-employed or employed individuals with uncertain incomes or incomes from multiple sources. These contracts may therefore also proxy this income uncertainty. There are also claims that self certification was used as an avoidance of due diligence and prudential lending; lending to households on social benefits, or with implausible income statements. In 2008 52% of all new mortgages were self certified (Financial Services Authority, 2010). 2 Rose controls for unobserved heterogeneity with the use of robust standard errors which allows for clustering in loans. This differs from our generalised modelling of unobserved heterogeneity and analysis of correlations between unobserved heterogeneity for the two forms of exit, and survival.

10 selectivity (i.e. the borrowers associated with a particular loan specie may share observed or

unobserved characteristics which may increase a specific risk). When several types of

mortgage are pooled in the same security then the heterogeneity of these specific risk may

multiply. The research reported in this paper addresses the selectivity that arises from the

individual specific factors that generate the initial choice of contract type by modelling these

factors as unobserved heterogeneity. We subsequently evaluate the effect of mortgage

contract choice upon both default and prepayment behaviour in a competing risk

framework.

3. ECONOMETRIC ESTIMATION

The use of econometric techniques in the mortgage loan literature has evolved from the use

of qualitative dependent variable models to the application of Cox Proportional Hazard

(CPH) and Multinomial Logit models (MNL) to incorporate competing risks. Further

developments have recognised the potential importance of unobserved heterogeneity,

particularly when modelling behaviour which from an option theoretic viewpoint appears

sub optimal (Deng et al, 2000, 2002). The MNL has the advantage over CPH that the

competing probabilities sum to unity modelling competing risk explicitly, because one risk is

at the expense of the other, but it has the disadvantage of assuming the independence of

irrelevant alternatives. There has been a tendency to favour the multinomial logit model

with modifications to allow for the correlation between specific unobserved components,

for example see Pennington-Cross and Chomsisengphet (2007).

The econometric methodology advances the literature in several ways. Firstly, we model

unobserved heterogeneity as a continuous distribution, rather than estimating parameters

for an arbitrary number of mass points that shift the base line hazard. Though there have

11 been some plausible a-priori categorisations of groupings of unobserved heterogeneity for

example: employment history; changes in marital status; household mobility (Clapp et al,

2006)- a specification that uses a more general form can cover a wide number of dimensions

(unobserved attributes and selectivity) and is more flexible. There are also likely to be

sources of selectivity bias in the choice of type of mortgage and other mortgage choices

made by households (Nichols et al, 2005). The econometric methodology treats unobserved

heterogeneity as a determinant for this selectivity.

The model presented here also allows for correlation between the sources of unobserved

heterogeneity that effect the various decisions (to remain current, to default or to prepay).

Though this has been used in a limited number of studies it is generally used in the context

of the Cox Proportional Hazard model (see Alexander et al, 2002). It has been suggested

that modelling unobserved heterogeneity with a more general functional form is too time

consuming/expensive and is not available in commercial software (Clapp et al, 2006).

Though the empirical models could be viewed as parsimonious this research speeds up

estimation and facilitates convergence of the likelihood function by using the methodology

of Train (2003) and the adaptation by Lanot (2008). For an outline of the estimation

procedure see Appendix A.

Overview

We wish to model the individual (discrete time) history of the decisions of default or early

repayment, { }1

iT

it td

= , jointly or conditionally on the history of a set of time dependent

regressors, say { }1

iT

it tx

= . Here we think of t as the history time and not the calendar time. iT

is the first of three possible times, either it is the date when the individual decides to repay

12 or decides to default, or it is the end of the observation period. itd can take three possible

values: 1 if the individual decides to keep paying the mortgage in period t, 2 if the individual

“decides” to default and 3 if the individual decides to repay the mortgage early.

Furthermore, we wish to account for the difference at the time of contracting between

mortgage/contract type and initial characteristics of the loan, say 0,i ic x with ic is

potentially a vector of qualitative variables indicating the type of contract chosen, while 0ix

measures the more quantitative aspects of the loan (the amount borrowed, the value of the

property on which the loan is based, etc). For example, it seems natural to distinguish

between certificated mortgage and self certificated mortgages and/or between fixed and

variable mortgages.

Finally we wish to account for unobserved differences between individuals which may affect

both ”exit” decisions independently or jointly. Hence we denote ( )1 2,i i iε ε ε≡ the vector of

unobserved individual factors. We assume that the marginal joint distribution of iε is

normal with means 0, unit variances and zero correlation. We discuss later on how this

specification captures the likely dependence between the two individual factors.

Assumptions

Firstly, we discuss the general assumptions we wish to maintain to estimate the parameters

of interest. In general, given a set of initial exogenous variables, say 0iw , we can always

express the probability density of a given history in the following general fashion; Equation

(1):

13 { } { }( )

( ) ( ) { }( ){ } { }( )

0 0 0 0 0

0 0

0 01 1

| 0 , | , 0 0 | , , , 0 01

| , , , , 0 01 1

, , , , |

| , | , | , , ,

| , , , , ,

i i

i

i i

T T

i i i it it it t

T

w i i c x w i i i i x c x w it i i i it

T T

d x c x w it it i i i it t

f c x x d w

f w f c x w f x c x w

f d x c x w

ε ε ε

ε

ε

ε ε ε

ε

= =

=

= =

=

(1)

This decomposes the joint density of all the quantities of interest into a product of marginal

and conditional densities. We denote by fs each density, and the subscript indicates the

identity (its argument and the set of conditioning variables) of each density.

For practical and computational reasons the restriction on the conditional distribution of the

time varying covariates represented by Equation (2) is maintained:

{ }( ) { }( )0 0 0 0| , , , 0 0 | , , 0 01 1| , , , | , ,i iT T

x c x w it i i i i x c x w it i i it tf x c x w f x c x wε ε

= == , (2)

All the information contained in the individual specific effect iε is captured by the the initial

value of the characteristics of the loan, 0ix , and the type of loan ic . This assumption is

plausible in fact since it suggests that the regressor’s history from time 1 onward is

independent from the individual specific effect conditional on the initial characteristics of

the loan 0,i ic x and the predetermined variables 0iw . This implies that we believe the

evolution of the time varying covariate is mostly determined outside of the individual time

invariant circumstances and/or decisions (this means that conditional on the choice of

interest rate the joint distribution of future interest rates is independent of individual

specific unobserved components).

14 Contract Choices, Selectivity and Unobserved Heterogeneity

Conditioning on the individual specific factors in the description of the distributions of the

initial characteristics of the individual loan contracts allows us to specify the dependence

between the initial characteristics and the outcome of interest, i.e. the timing and the

nature of the exit decisions. This provides a “natural” parametrisation for the (dynamic)

selection effects we would expect to observe. In principle we are able to evaluate quantities

such as the distribution of the duration until defaults/repayment given the observed initial

characteristics or the distribution of the expected time to default/repayment given the

observed initial characteristics. The introduction of the unobserved component allow for a

more “flexible” correlation structure between the diverse elements of the models. In

particular it is more flexible than a comparable specification (with the factor loadings for the

individual specific effect set to 0) which would rely only on conditional independence.

The density of the initial observed characteristics of the loan is assumed to be jointly

normally distributed with a mean vector depending linearly on iε and a constant variance

covariance (although it would be feasible to make the variance covariance matrix

dependent on some exogenous observed characteristics as well as dependent on the

individual specific effects in iε ). This assumption is formally presented by Expression (3).

( )0 0 00 0 0 1 1 2 2| , ~ ,i i i i i ix w N wε ε εΚ + ∆ + ∆ Σ , (3)

where 0Κ is a parameter matrix and 01∆ and 0

2∆ are parameters vector conformable to the

dimensions of 0ix . Σ is the variance-covariance matrix for 0ix given 0iw and iε .

15 We assume that the probability density of the mortgage type is of the logit or multinomial

logit form in the various dimensions of choice (i.e. certification on the one hand and interest

rate choice); Equations (4) and (5). Firstly, in the case of the certification choice we assume

that the probability is:

( ) ( )( )1 0 0| , , 0 0 1 1 1 10 0 0 0 1 1 2 2

10 | , ,

1 expc x w i i i

i i i i

f x wx w

ε εα κ δ ε δ ε

=+ − + + +

, (4)

( ) ( )1 0 0 1 0 0| , , 0 0 | , , 0 01 | , , 1 0 | , ,c x w i i i c x w i i if x w f x wε εε ε= − . (5)

Since there are three potential choices for the interest rate choice (fixed, discount or Libor),

given the certification choice we assume that probabilities take the form of Equation (6).

( )( )

( )

2

2 1 0| , , 2 1 0 0 3

1

| , , ,i

i

Z c

c c x i i i iZ k

k

ef c c x w

eε ε

=

=∑

, with c2 in {1,2,3}, (6)

Where:

( )( )( )

2 2 2 2 22 1 0 2 0 2 21 1 22 2

2 2 2 2 23 1 0 3 0 3 31 1 32 2

1 0,

2 ,

3 .

i

i i i i i i

i i i i i i

Z

Z c x w

Z c x w

λ α κ δ ε δ ε

λ α κ δ ε δ ε

=

= + + + +

= + + + +

In some cases one of the three options (option 3) is not available at a particular time, and

the model above collapses to a simple binary logit model.

16 Estimation

Equation (7) represents the assumption that the conditional joint likelihood of the history of

decisions can be decomposed into a product of conditional probabilities:

{ } { }( ) { }( )( )

0 0 0 0

0 0

| , , , , 0 0 | , , , , 0 01 1 11

| , , , , 0 01

| , , , , | , , , , ,

| , , , , , .

ii i i

i

TT T T

d x c x w it it i i i i d x c x w it it i i i it t tt

T

d x c x w it it i i i it

f d x c x w f d x c x w t

f d x c x w t

ε ε

ε

ε ε

ε

= = ==

=

=

=

∏

∏ (7)

Practically we assume that each probability ( )0 0| , , , , 0 0| , , , , ,d x c x w it it i i i if d x c x w tε ε is of the

multinomial logit form; Equation (8):

( )( )

( )0 0| , , , , 0 0 3

1

| , , , , ,it

it

V d

d x c x w it i i i iV k

k

ef d x c x w t

eε ε

=

=∑

, with d in {1,2,3},

(8)

where:

( )( )( )

3 3 3 3 3 32 2 2 2 1 0 2 0 2 21 1

3 3 3 3 3 3 33 2 3 2 1 0 3 0 3 31 1 32 2

1 0,

2 ,

3 .

it

it it i i i i i

it it i i i i i i

V

V x c c x w

V x c c x w

β γ λ α κ δ ε

β γ λ α κ δ ε δ ε

=

= + + + + +

= + + + + + +

The parameters of interest which appear in the conditional probabilities are therefore

( )0 1 1 3 3 0 00 0 1 2 1 2, , , , , , , , , , , ,j j j j j

k k k k k k kα α λ κ κ β γ δ δΚ Σ ∆ ∆ with j=2,3 and k=2,3. The parameters

0 01 2 1 2( , , , )j j

k kδ δ∆ ∆ are the loadings of the individual specific component in the conditional

densities/probabilities and capture the dependence between the dependent variables and

the unobserved individual specific unobserved components.

17 Of particular interest is the interpretation of the sign and magnitude of the parameters

3 3 321 31 32, ,δ δ δ in the conditional density of the repayment/default decisions. Recall that we

assumed that the components of iε are uncorrelated; this is however only a matter of

presentation. Indeed, we are always able to define the parameters 331δ and 3

32δ as functions

of an unrestricted parameter 3δ and a correlation coefficient ρ as shown by Equation (9):

3 331δ δ ρ= and ( )1/23 3 2

32 1δ δ ρ= − . (9)

The term 3 331 1 32 2i iδ ε δ ε+ can then be written as ( ){ }1/23 2

2 11i iδ ε ρ ρε− + and the term in

braces is then a normal variate correlated, by construction, with 1iε . We therefore have the

correspondence:

3 3 331 32

3 3 331 32

3 3 331 32

3 3 331 32

0, 0 0, 0;

0, 0 0, 0;

0, 0 0, 0;

0, 0 0, 0.

δ δ δ ρδ δ δ ρδ δ δ ρδ δ δ ρ

> > ⇔ > >

> < ⇔ < <

< > ⇔ > <

< < ⇔ < >

The ratio of 3 331 32δ δ is an increasing function of ρ only and therefore its inversion gives an

estimate of ρ , and given this estimate it is then straightforward to obtain an estimate for

the parameter 3δ (applying the delta method to the transformation given the precision for

331δ and 3

32δ will provide an easy way to obtain the precision for ρ and 3δ ).

The difficulty with the estimation of these kind of models resides in the fact that the

individual specific effect are not observed, and therefore the observed likelihood is derived

from the latent likelihood described above by integrating out the individual specific effects.

This observed likelihood is potentially difficult to evaluate (and therefore optimise), since it

18 requires to integrate a product of terms over the multidimensional density of the

unobserved component. Instead of taking this direct route, we adapt the EM algorithm

described in Train (2008), which is used in the case of the estimation of discrete mixtures to

the case where the distribution of the unobserved component is known in order to obtain

the maximum likelihood estimates. The advantage rests in the fact that both the E-step and

the M-step are relatively straightforward in our case, and that in particular the M-step only

requires the maximisation of the sum of standard (concave) logit likelihoods and of the

likelihood for a multivariate normal variable, weighted by quantities which are directly

calculated from the usual normal quadrature abscises and weights and the complete latent

likelihood (see the appendix for some details). The precision of our estimates is calculated

from the latent likelihood used in the EM algorithm using the results derived in Oakes

(1982) and adapted to this case in Lanot (2003).

4. DATA AND EMPIRICAL MODEL

The Data

The data covers 100,000 mortgage contracts, and when constructed as a panel contains

approximately two million observations (individual x time points). The mortgages were

issued by a single US originator operating in the UK non-conforming residential mortgage

market, but the pools also contain some prime and near prime debt. These issues remained

the property of a major global investment bank with which one of the researchers

undertook collaborative work. The research is subject to confidentiality agreements, and as

such the identity of the data source cannot be disclosed3. 3 Confidentiality is maintained by not estimating models on particular tranches of securitized debt, but rather incorporating the whole issue for analysis. Though much of the data is now in the public domain the absence

19 The mortgages are classified by issue and there were two issues per year from January 2003

to May 2006, offering eight issues over four years (labelled 103, 203, 104, 204, 105, 205,

106, and 206). The data was collected by issue rather than by the tranches of loans

packaged into different securities. For example, the January 2003 issue (103) is a mortgage

portfolio available in eight tranches but we were not given the means to identify these. The

data covers the full range of types of loan that were securitized, including fixed rate,

discounted interest rates, buy to let and self certified mortgages.

The data was reduced by removing the comparatively small number of prime and near

prime loans (0.0789 and 0.0432 of total observations respectively) concentrated in

particular issues. Buy to let mortgages (0.0842 of observations but with 0.508 of buy to let

also prime loans) were also excluded so that the sample included only first lien loans on

owner occupied property. Given the magnitude of the data set, and the difficulties of

achieving convergence with the econometric models used, the data was broken down into

four sub samples. Sample 03 contains issues 103 and 203, those for 2004 are in sample 04,

2005 sample 05 and 2006 sample 06. Using these year by year samples assisted estimation

and allowed a comparison of changes in parameter estimates and unobserved

heterogeneity for mortgages issued and pooled over different periods of time.

Reference to the descriptive statistics by sample shown in Table 1 reveals that there is

significant variation in the representation of different types of contract across and within

issues. In particular the number of self-certified mortgages is significantly higher in the 04

sample. The proportion of fixed rate mortgages also increases in samples 05 and 06

compared to 03 and 04. Variations in contract terms increased significantly for later issues of dates of redemption and repossession on investor reports means that the timing of exits from the pool cannot be reconstructed. Therefore public domain data is not fully useable beyond May 2008.

20 (samples); for example issue 203 has 49 different contracts and by issue 106 this has

increased to 639. Insofar as different types of mortgage exhibit differences in loan

performance then mortgage pools and their tranches of securities will also differ. The

incidence of types of contract, across the issues/samples, led to low representation of some

contracts, or co linearity problems such that not all contract choices could be modelled with

each issue. This was not a problem as each contract choice could be modelled within at least

one sample.

The investment bank selected the loans of this particular mortgage originator for analysis.

Therefore there is concern that the pools may be subject to selectivity bias. Each of the

issues used in the research originally had a triple A credit rating from Standard and Poor and

Moody’s, with an Aaa from Fitch. By March 2009 the ratings of some of the issues had fallen

to BBB and BBB-. Though the data was from an originator in the top decile of average loan

performance there was considerable variation by issue and between the yearly grouping of

issues. This variation in performance can be seen in Figure 1 and Figure2 which show

conditional prepayment and conditional default rates by issue. It is also the case that the

legitimacy of credit ratings of AAA rated debt can be disputed (Lupica, 2008; Rosengren,

2010).

Insert Figure 1 and Figure 2

Conditional Prepayment Rates and Conditional Default Rates by Issue

We compared the default rates observed in our data with the mean and standard deviations

of monthly repossession rates for 160 issues from 26 mortgage originators, evaluated by the

same credit rating agency as the pools studied here. The comparison covered a period of

twenty four months since the mortgages were pooled. The mean value for all issues was

21 0.81% with a standard deviation of 1.15%. The mean values for all the samples were within

one standard deviation of the overall mean (0.60, 0.29, 0.75, and 0.70 for sample 03 to

sample 06 respectively). Earlier dated pools performed better than those issued at a later

date introducing some significant variation. Similar results were obtained for comparisons

made at twelve and eighteen months. We conclude that the issues used in estimation can

be taken as a “representative” sample, though not indicative of the worst performing

subprime pools.

Variable Definitions and Measures

Empirically it is important to assess how far the call option to prepay and the put option to

default are ‘in the money’. Given that the value of embedded options is a complex function

of stochastic variables then it is difficult to measure precisely the intrinsic value of an

option; and so ‘indirect’ measures are used to evaluate the likelihood of the call or put

option being ’in the money’. We follow Pennington-Cross and Chomsisengphet (2007), and

use an estimate of the current loan to value ratio (currentlv) on a mortgage holder’s

property to represent the extent to which the put option is ‘in the money’. The more likely

that the put option to default is ‘in the money’ (negative loan to value ratios) the higher the

probability of a household defaulting on the mortgage debt. The descriptive statistics for

this and other variables are given in Table 1.

Insert Table 1

To indicate the extent to which the call option is ‘in the money’ we again follow Pennington-

Cross and Chomsisengphet and use the change in interest rates since the date of origination

(libor change). Given that the typical index rate for subprime mortgages is 3 month Libor

we use the Libor index as the representative rate. For the UK it is expected that

22 endogenously determined financial behaviour is more likely in the case of prepayment than

default, so as a further measure of the value of a call option to prepay the standard

deviation of Libor (stdlibor-a moving standard deviation over 12 months) is included as an

independent variable4. It is expected that the call option to prepay has a higher value when

interest rate volatility is high and therefore there is an incentive to keep the option, the

likelihood of prepayment then being less.

The empirical specification also includes variables that represent exogenous payment

shocks that might influence default and prepayment. One measure of this is the interest

rate shock, which is the change in the actual interest rates paid since the date of origination

of the contract (actual_shock). There is no data on incomes which can be used with

mortgage payments to represent the ability to pay. Given the absence of such data then the

interest rate shock is used as a proxy for this, with larger shocks being more likely to lead to

difficulty in paying. This may then lead to default, or prepayment to seek out discounts and

cheaper rates. We also control for the general level of interest rates by including the current

level of Libor (libor).

The data set does not contain indicators of personal characteristics, and to some extent the

influence of these variables is attributed to unobserved heterogeneity in the samples. One

variable that may proxy credit worthiness is the original loan to value ratio (Pennington-

Cross and Chomsisengphet, 2007). This may also proxy the nature of the household balance

sheet and its riskiness (Harrison et al, 2004). Though some studies have included both the

current loan to value ratio and the original level of gearing in the same specification 4 The period exhibits a continuous increase in nominal house prices and low levels of volatility and negative equity (stats). For this and the other reasons stated in the text exercising the option to default is considered less likely than exercising the option to prepay.

23 (Pennington-Cross and Chomsisengphet, 2007), a high degree of correlation between these

two measures can be problematic. The research reported here addresses this problem by

including the initial size of mortgage (log initial loan balance) and the value of the property

at the date of origination of the mortgage (log initial house value) as independent variables.

Purchasing a more expensive property and having a larger absolute size of loan is also a

means of overcoming the prudential lending constraint of a maximum loan to value ratio

(Brueckner, 2004) and may therefore increase the risk of facing payment difficulties at some

point of time.

The key characteristics of mortgage contract design are indicated by dummy variables. Thus

we have indicator variables to represent the choice of a fixed rate mortgage (fixed=1), and a

discounted mortgage (discounted=1); mortgages with the standard variable rate or Libor are

excluded for identification. Self-certification is also indicated by a categorical variable

(selfcert=1). The review of previous research suggested that the sign on discount would be

positive for both default and prepayment with the estimated parameter on discount being

larger than that for fixed. Self- certification is taken as an indicator of information

asymmetry and adverse selection and selfcert=1 is expected to lead to higher defaults and

prepayments.

A further significant feature of mortgage design is the existence of the interest reset date on

which discounts, or periods during which the rate of interest has been fixed, end. A dummy

variable (revert) is used to represent the current mortgage month if it is within the period

prior to the reset date (revert=1). Following (Ambrose et al, 2005) it is expected that both

defaults and prepayments will be higher in the post reversion period. Defaults may increase

because the increase in the interest rate leads to a payment shock that was not fully

24 considered when the mortgage was taken out, that is consumers may have been myopic

(Miles, 2004). Prepayments may increase because post reversion the call option is more

likely to be ‘in the money’, or households may simply be augmenting their cash flow by

seeking cheaper alternative mortgage deals.

5. EMPIRICAL RESULTS

The estimates for most parameters show consistent sign and magnitude across the four

samples5. Irregular cases are discussed later on. We look first of all at the parameter

estimates for those variables which identify features of mortgage contract design and

discuss these separately for default and then refinancing behaviour. The results are then

evaluated with respect to the time varying variables that largely represent the options

embedded in the mortgage and/or reflect exogenous payment shocks and affordability. The

parameter estimates for each of the four samples are reported in Table 2. The estimates

represent shifts in the baseline hazard which is measured by time out from the date of

origination of a mortgage (log of months).

Insert Table 2

Mortgage Contract Design

With respect to mortgage contract design the likelihood of default is increased by self-

certification (selfcert=1). However, a discounted mortgage (discount=1) reduces the

likelihood of default. There are no statistically significant effects on early repayment or

default for holding a fixed rate mortgage (fixed=1). The sign and significance of selfcert is

consistent with higher default rates. The information problems and adverse selection that 5 For estimation purposes the data has been standardised with mean zero and a standard deviation of one.

25 accompany the use of self-certified mortgages are a likely explanation for this observed

pattern. The lower likelihood of default that accompanies holding a discounted mortgage

reflects the favourable impact of teaser rates on affordability.

Default is found to be more likely the larger the original loan (log initial loan balance) and

the lower the original house price (log of initial house value); these two variables have the

largest parameters (5.1186 and 4.2209 respectively in Sample 03). While larger loans might

reflect a good credit rating they also bear higher servicing costs and this may explain the

higher likelihood of default indicated by our estimates. A low purchase price for a property

(low value) may reflect other factors such as occupational status and wealth, but also

indicates the possibility of less absolute value to use as collateral for further borrowing;

liquidity constrained households with little or no equity in their property may have their

borrowing constrained in the non housing loan market.

The likelihood of prepayment is increased when selfcert=1 and with discount=1 . One

possible explanation is that the higher rates of interest on self-certification may lead to

risky borrowers, who do not disclose their incomes and who gradually repair their credit

records, eventually seeking less expensive deals in the prime lending sector, or with a new

subprime lender. There is also significant selectivity attached with those choosing

discounted mortgages. Holders of discounted mortgages who may be cash constrained may

have a greater incentive to shop around for other teaser rates.

The likelihood of refinancing is also increased when log initial loan balance is large, and

when log initial house value is low; once again these exhibit the largest parameter values

(0.4662 and 0.6191 respectively in Sample 03). Mortgage holders with larger loans can make

higher absolute savings from searching for new mortgage deals. Households with lower

26 valued houses may have less equity than those with higher valued properties, and therefore

seek the release of cash through seeking more competitive mortgage deals, rather than

through further borrowing. Again, the results are compatible with an emphasis by

households upon cash flow and affordability.

The existence of prepayment penalties and a date at which the favourable contract rates

revert back to the higher index interest rate are other important features of mortgage

contract design. The dummy for post reversion decisions (revert=1) was not statistically

significant in the default equations. Thus the increase in mortgage payment did not induce

default. This results in part because of the increase likelihood in the competing risk, i.e.

prepayment. The positive and significant coefficient on revert in the prepayment equations

signals the possibility that an increase in the required mortgage payment induced re-

contracting.

Insert Figures 3.a to 3.d

Hazard and Sub Hazards by self certification and reversion=1

Figures 3.a and 3.b illustrate the effects of self cert upon the hazard rates and sub hazard

rates for default (3.a) and prepayment (3.b). The estimates are based on the characteristics

of the longest surviving observation that can be found in Sample 03. The simulations plot

the reaction to the path of time varying variables such as Libor and are based upon the

choice of a fixed rate mortgage (Fixed=1) prior to reversion. The hazard and sub hazard

curves for default are shifted markedly upwards by the presence of self certification and

significantly but less so for prepayment.

Figures 3.a and 3.b incorporate two alternative ways of measuring the risk in the population

of exiting at a point of time t. The Hazard schedule depicts the ‘cause specific hazard’ where

27 the population at risk (the risk set) contains survivors from all causes of exit up to a given

time t. The sub-hazard differs from the hazard in so far as the relevant risk set includes

those who have exited using any exit route other than the one being analysed, in addition to

the survivors up to time t. For example, the sub hazard for default will have those

individuals who prepaid their mortgage included in the risk set (see Fine and Grey, 1999;

Lau, Cole and Gange, 2009). Hence the cause specific hazard measures the rate of exit at

time t for a given cause given survival so far, while the sub-distribution hazard measures the

exit rate at time t, but conditions both on survival up to time t and on the possibility that

some individuals who have not survived up to t may have been at risk of the specific cause

at time t. Qualitatively the impact of self certification is the same for prepayment and

default though in the case of default the sub hazard schedules are much lower than the

hazards and the rate of exit through default now fall away after twenty eight months.

Option Theoretic and Affordability Considerations

The current (i.e. measured at time t) loan to value ratio (currentlv) is used to proxy the

extent to which the put option to default is ‘in the money’ with the expectation is that its

associated parameter will have a negative sign. For default this variable is not statistically

significant at the 5% level in three of the samples. Given, that UK mortgages are debt with

recourse then there is less likelihood of observing ruthless default in the United Kingdom

than in the United States mortgage market. Other parameter estimates suggest that

affordability may be a more critical issue for default than endogenous financial calculation.

For example, the extent to which interest payments changed since origination of the

mortgage contract (actual_ shock) has a positive and statistically significant effect on the

likelihood of default.

28 Change in Libor since origination is the proxy for the extent to which the call option to

prepay is ‘in the money’. The parameter estimates for this variable (liborchange) are not

consistent across the four samples. The earlier samples Sample 03 and Sample 04 do have a

statistically significant negative sign but has a positive effect in the two subsequent samples.

The variable stdlibor provides a further test of the option theoretic explanation of

refinancing behaviour. In this case the expected negative sign is found in Samples 05 and

Sample 06 but the parameters are positive and statistically significant in Sample 03 and

Sample 04. Thus the results for testing the option theoretic explanation of prepayment

behaviour are ambiguous. The typically larger coefficient on actual_shock across samples,

compared to that for libor change ( 0.7721 and -0.1858 respectively in Sample 03) and the

lack of statistical significance of libor change again suggests the importance of affordability.

There are also possible exogenous shocks on prepayment behaviour. Large payment shocks

may induce cash constrained borrowers to seek better deals. The change in interest

payments on the mortgage since the origination of the debt (actual_shock) has a positive

and statistically significant effect upon the likelihood of prepayment.

Those results where the estimates are inconsistent in sign across samples involve time

dependent variables, in particular those variables reflecting upon the option theoretic

interpretation of household mortgage choices. The inconsistency may be the result of

complex interactions with mortgage contract features for which we do not control;

alternatively households and/or credit market conditions may differ across the samples.

Credit market conditions deteriorated throughout 2007 and up to mid 2008. During this

period Libor increased markedly, and with it mortgage rates, reflecting changed market

perceptions of risk. Volatility in this case is likely to indicate the several rises in Libor which

29 corresponded with a marked reduction in contracts available for refinancing. Though the

four samples all have lives which extend into 2008, later issues have more mortgages that

are likely to be affected by these changes. Samples 05 and 06 have a positive sign on libor

change suggesting that higher interest rates induced a search for contracts with lower rates

to minimise payments. There is a negative sign on stdlibor implying that tightening credit

market conditions reduced prepayments, possibly through less competitive deals being

available and/or access to mortgage finance being rationed. These results suggest the

importance of affordability and ease of access to mortgage finance driving prepayment

decisions6.

Table 3 summarises the observed signs on the estimates of unobserved heterogeneity and

the implied correlation between these estimates for each equation (B1D, B1R, B2R). The

first point to note is the consistency of sign across the four samples. The sign on B2R implies

a negative correlation between the unobserved components of default and prepayment so

that a reduction in the likelihood of prepayment increases the likelihood of default. This is

compatible with the perception that the latter part of 2008 stopped credit impaired

households from improving their mortgage terms and thus increasing the risk of

delinquency and mortgage default.

Insert Table 3

6 Comparisons were made between simple logit estimates using the type of exit as the dependent variable with a logit with unobserved heterogeneity. There were no significant changes required in the interpretation of parameter estimates though controlling for unobserved heterogeneity did increase the size of parameter estimates.

30 6. CONCLUSION

This paper considered whether the variety of mortgage contract designs that were

securitized explains the performance of subprime securities, and their supposed

idiosyncratic behaviour. A model was estimated based upon the competing risk of mortgage

defaults and prepayments controlling for individual unobserved heterogeneity. The

unobserved individual specific factors were modelled in a flexible general form, allowing for

their influence upon the initial choice of contract type (for example self certification;

discounted; fixed). The mixing of a large number of different types of contract into pools of

securitized subprime loans may be one reason for their supposed wide variability in risk and

return, and presumed unpredictability. The pools of mortgages had different proportions of

contract type and an increasing variety of contract terms. Without any assessment of the

impact of these contract variations, particularly self-certified/low or zero documentation

mortgages, it is not surprising that performance might be viewed as ‘idiosyncratic’.

The significance and interpretation of the impact of mortgage contract design on loan

performance depends upon whether mortgage choices can be seen as the exercise of

embedded call and put options, or as empirically determined by exogenous shocks and

factors influencing affordability. The estimation suggested that treating mortgages as

embedded option contracts did not explain the default and prepayment behaviour of the

sample. Given this then the impact of reversion periods, the information asymmetry and

adverse selection associated with self-certification- and only fixing interest rates on

mortgage contracts for short periods- largely operated though their impact upon

affordability. These affordability issues resulted in adverse effects on default and generated

highly active amounts of prepayment for periods where contract choices were plentiful.

31 There is little evidence, in the samples analysed here, of significant variations in unobserved

heterogeneity between pools of mortgages. The main differences in mean loan performance

between securities are most likely to arise from compositional effects resulting from having

different proportions of contracts with different features. There was an observed change in

the behaviour of later pools of debt, possibly arising from the change in credit market

conditions which restricted refinancing by liquidity constrained households. To the extent

that these considerations and unobserved heterogeneity were not taken into account when

pricing securitized bonds then the behaviour of these pools would appear idiosyncratic.

Thus securitized subprime loans may be given meaningful valuations on bank a balance

sheet; that is if the behaviour resulting from the mix of mortgage designs used in the

securities is better understood.

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Appendix

In this appendix we describe the E-step and the M-step of the optimisation algorithm we

use to estimate the parameters of the model.

To simplify this presentation we express the complete latent likelihood for the ith

observation as

( ) ( ) ( ) ( )1 2 1 2L , ,i iLθ θ ε ε φ ε φ ε= , A.1

where ( )1 2, ,iL θ ε ε is the likelihood of observation i given ( )1 2,ε ε and where ( )φ ε is the

normal density function. In principle the observed likelihood is deduced from A.1 by

integrating the complete latent likelihood over the range of ( )1 2,ε ε :

( ) ( ) ( ) ( )1 2 1 2 1 2, ,i iL L d dθ θ ε ε φ ε φ ε ε ε= ∫∫ A.2

Instead of considering the continuum of possible values for ( )1 2,ε ε we limit ourselves to H

values for iε , say { }1 2, , ..., He e e and we associate with each couple ( )',h he e a positive weight

( ), 'p h h such that ( )1 ' 1

, ' 1H H

h h

p h h= =

=∑∑ , and such that ( )1 ' 1

, ' 0H H

hh h

e p h h= =

=∑∑ ,

( )2

1 ' 1

, ' 1H H

hh h

e p h h= =

=∑∑ and ( )'1 ' 1

, ' 0H H

h hh h

e e p h h= =

=∑∑ . We deduce such the values { }1 2, , ..., He e e

and the weights ( ), 'p h h from the Gauss-Hermite quadrature abscissas and weights for a

given H (see Press et al., 1986, for an introduction and Abramowitz and Stegun, 1964). We

therefore approximate A.1 with

( ) ( ) ( ) ( )( ) ( )

( ) ( )( ) ( )

1 2 1 2

'

. '

'1...

' 1...

L , ,

, , , '

, , , 'i

i i

i h h

h h

i h hh Hh H

L

L p h h

L p h hδ

θ θ ε ε φ ε φ εθ ε ε

θ ε ε==

=

≈

= ∏

A.3

for some h and h’, where we define ( ), 'i h hδ to be equal to 1 if observation i is of type

(h,h’) and 0 otherwise. In effect we augment the observed data with ( ), 'i h hδ the (latent)

“type“ of each individual observation. From A.3, the (approximate) latent log-likelihood can

now be written as

( ) ( ) ( ) ( ){ }'1...

' 1...

lnL , ' ln , , ln , 'i i i h hh Hh H

h h L p h hθ δ θ ε ε==

≈ +∑ A.4

36 For some values for the parameters, say χ , the EM algorithm process first (Expectation

step) by calculating the Expected latent log-likelihood given what is observed (which we

represent by Oi) ( ) ( )( )

( ) ( ) ( ){ }'1...

' 1...

E lnL | ln ;

E , ' | ln , , ln , '

i i

i i i h hh Hh H

O L

h h O L p h h

χ

χ

θ θ χ

δ θ ε ε==

≡ ≈ + ∑ A.5

where [ ]E | iZ Oχ evaluates the conditional expectation of a random variable Z given Oi and

evaluated with the distribution parametrised by the vector χ . The key to A.5 is the fact that

( ) ( ) ( ) ( )' 'E , ' ln , , | E , ' | ln , ,i i h h i i i i h hh h L O h h O Lχ χδ θ ε ε δ θ ε ε= (given the type (h,h’)

the value of the log-likelihood ( )1 2, ,iL θ ε ε is constant given Oi. In the present case since we

specify that ( )φ ε is the standard normal distribution, the weights ( ), 'p h h and abscissas

{ }1 2, , ..., He e e are known (for any given H) and don’t need to be estimated (this would be

the case if instead we assumed that the joint distribution ( )1 2,ε ε was unknown. Then the

present EM algorithm could be amended to estimate this unknown joint distribution).

Hence we can evaluate ( )E , ' |i ih h Oχ δ from the value of ( )'ln , ,i h hL θ ε ε for all types

( ) ( ) ( )( ) ( ) ( )'

', '

, , , 'E , ' | , ';

, , , 'i h h

i i ii k k

k k

L p h hh h O h h

L p k kχχ ε ε

δ π χχ ε ε

= ≡ ∑

A.6

The second stage (Maximisation Step) we maximise with respect to θ the Expected latent

log-likelihood given what is observed and given some initial value χ for the parameters.

This procedure is repeated until convergence, where θ becomes the next value for χ , that

is until

( ) ( ) ( ){ }'1 1...

' 1...

=arg max , '; ln , , ln , 'N

i i h hi h H

h H

h h L p h hθ

χ π χ θ ε ε= =

=

+∑ ∑ . A.7

The EM algorithm has good properties and if it converges it an be shown that it produces

the maximum likelihood estimator (see Gouriéroux & Monfort,. 1995). The benefit of using

the EM algorithm arises in practice since the objective in A.7 can be understood as the

maximum likelihood based on the latent log-likelihood but weighted by the quantities

( ), ';i h hπ χ (which are treated as given within each M-step).

37 In our context the latent log-likelihood can be decomposed into the sum of several terms

each involving a different set of parameters. Hence the M-step is obtained by the separate

maximisation of each of the “independent” components of the properly weighted latent

log-likelihood. To illustrate this property assume that we can write:

( ) ( ) ( )1 1 2 2' ' 'ln , , ln , , ln , ,i h h i h h i h hL L Lθ ε ε θ ε ε θ ε ε= + , A.8

with ( )1 2,θ θ θ= with 1θ and 2θ distinct. Then the M-step is amounts to two separate

maximisations

( ) ( ) ( ){ }1

1 1 1'

1 1...' 1...

=arg max , '; ln , , ln , 'N

M i i h hi h H

h H

h h L p h hθ

θ π χ θ ε ε= =

=

+∑ ∑ ,

( ) ( ) ( ){ }2

2 2 2'

1 1...' 1...

=arg max , '; ln , , ln , 'N

M i i h hi h H

h H

h h L p h hθ

θ π χ θ ε ε= =

=

+∑ ∑ .

38 Table 1 Descriptive Statistics Samples One to Four Sample 03 n=497570 Sample 04 n=515654 Variable Mean Std. Dev. Mean Std. Dev Exit 2.971532 0.1693367 2.971211 .1728018 months 2 0.30177 14.45543 18.74752 12.66308 loantovalue 0.6955786 0.147224 0 .6858394 0.1445353 selfcert 0.5765159 0.494111 0 .7476021 0 .4343887 currentlv 0.5311837 0.153348 0 .5505707 0 .1462581 actual_shock 1.133196 1.085864 1.373973 0 .9628001 discount 0.8709388 0.335268 0.8820352 0.3225667 fixed 0.1234459 0.328947 0.1169214 0.3213269 currentlibor 4.313035 0.581544 4.817942 0 .4355623 liborchange 0.385928 0.621803 0.8091501 0.4684648 revert 0.3794903 0.485266 0 .4077017 0.4914077 stdlibor 0.2107883 0.1059696 0 .2403952 0 .1243599

39 Table 1 cont. Descriptive Statistics Samples One to Four (cont.) Sample 05 n=455826 Sample 06 n=465386 Variable Mean Std. Dev Mean Std. Dev Exit 2.974975 .1650379 2.981961 0.1412379 months 19.49204 15.99471 17.92071 15.84958 loantovalue 0.6991869 0.14986 0.7314664 0.164162 selfcert 0 .6167397 0.4861814 0.5950136 0.4908899 currentlv 0 .584231 0.1713933 0.5975107 0.1858507 actual_shock 0.5861474 1.060004 0.2322083 0.7776358 discount 0.4997236 0.5000005 0.3423244 0.474488 fixed 0.3742437 0.4839275 0.6202851 0.4853164 currentlibor 4.95956 0.5164024 5.168357 0.6255065 liborchange 0.0623435 0.5970122 0.4107663 0.8862877 revert 0.4647256 0.4987547 0.7453705 0.4356532 stdlibor 0.2083212 0.1276791 0.2629969 0.1319354

40

Table 2

Parameter Estimates: Competing Risk Model With Unobserved Heterogeneity Sample 03 Sample 04 Sample 05 Sample 06

Default Equation Parameter Std

Error Parameter St

Error Parameter Std

Error Parameter Std

Error constant

-9.9636 0.8048

-8.8620 0.2431

-10.7801 0.8021

-10.3914 0.8961

log months 2.1515

0.6026

2.2422

0.2299

3.9253

0.4897

2.9253

0.4661

actual_shock 0.7721

0.2914

0.5830

0.1610

0.3060

0.1241

0.5327

0.0708

currentlv -0.2136 0.2179

0.1146 0.1366

-0.0566 0.1775

0.7016 0.2104

libor -0.1257 0.3565

0.2205 0.1582

-1.4480 0.2885

-2.7527 0.4377

libor change -0.1858 0.3153

-0.6101 0.1566

1.2295 0.3082

3.5136 0.5603

revert 0.0613 0.1085

-0.0178 0.0885

-0.1121 0.0619

0.0253 0.0355

stdlibor -0.0172

0.0803

-0.1426

0.0965

-0.1865

0.1092

0.1116

0.1688

self cert 0.3800

0.0934

- - 0.2898

0.0595

0.3323

0.0762

discount

-0.3218

0.1531

-0.1164

0.0800

-0.4626

0.0970

fixed 0.0725

0.0984

libor mortgage 0.6211

0.1366

-0.2993

0.1147

Log initial loan balance

5.1186

0.9106

4.0890

0.5342

4.5993

0.6222

2.7639

0.6470

Log initial house value

-4.2209

0.7502

-3.4524

0.4365

-3.6623

0.4899

-2.4259

0.5387

Repayment Equation constant

-4.2544

0.0426 -4.6005

0.0790

-4.6542

0.0387

-6.1337

0.1616

log months 2.0366

0.0947

1.6595

0.1154

1.7099

0.0562

2.2951

0.1395

actual_shock 0.3351

0.0354

0.6032

0.0427

0.0611

0.0267

0.8050

0.0475

currentlv -0.0364

0.0523

-0.2094

0.0549

-0.1502

0.0406

-0.2461

0.0659

libor -0.4951

0.0664

0.5855

0.0448

-0.3255

0.0483

-2.0097

0.1414

libor change -0.0318

0.0621

-0.5707

0.0535

0.4871

0.0487

2.4183

0.1572

revert 0.0866

0.0107

0.1376

0.0097

0.2044

0.0076

0.4096

0.0073

stdlibor 0.1010

0.0123

0.0450

0.0195

-0.1090

0.0261

-0.1587

0.0392

self cert 0.1073

0.0172

- - 0.0307

0.0135

0.1952

0.0266

discount

-0.1804

0.0389

0.1104

0.0329

-0.6902

0.0474

fixed 0.0906

0.0238

libor mortgage -0.0291

0.0500

-0.3924

0.0447

log initial loan balance

0.4662

0.2595

0.6105

0.1396

0.3677

0.0860

0.0029

0.1527

log initial house value

-0.6191

0.1796

-0.6352

0.1089

-0.2445

0.0608

-0.2118

0.0966

Unobserved Heterogeneity B1D

-2.0668

0.5984

-1.0956

0.2823

-2.4980

0.4252

-2.1758

0.4766

B1R

-0.7609

0.1633

-1.1409

0.1081

-0.7120

0.0792

-1.1584

0.1519

B2R 0.8974 0.1590 1.4385

0.1106

0.3908

0.1007

1.3184

0.1809

41 Table 3 Parameter Estimates: Unobserved Heterogeneity Pool One Pool Two Pool Three Pool Four B1D <0 <0 <0 <0 B1R <0 <0 <0 <0 B2R >0 >0 >0 >0 r <0 <0 <0 <0

42

43